You and a friend are playing a game. Between you is a pile of 15 pennies. You’ll take turns removing pennies from the pile — each of you, on his turn, can choose to remove 1, 2, or 3 pennies. The loser is the one who removes the last penny.
You go first. How should you play?
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Consider what happens if you leave your opponent 5 pennies. At that point, no matter how many he removes, you can play so as to leave the final penny to him. Now, if leaving him 5 pennies is a worthy target, then by the same principle so is leaving him 9 pennies: No matter how he plays, he must now give you the opportunity to leave him with 5. And going one step further, aiming for 13 pennies will ensure that you can reach 9, then 5, and victory. So, on your first turn, take 2 pennies.
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